Would there be any reason why your daughter would be upset when you are out during the day,did anything happen in the time frame , could she have heard some thing regarding parents leaving or dying, any adult conversations overheard , sometimes Children get things out of proportion ,they see or overhear problems and its possible you have been unaware of,,the very fact she tells you to be safe, she is concerned about you, can you think why she would ask you to be safe. Could she have seen a Movie that what scary regarding a Mothers Loss? I hope all is well.
This could be separation anxiety - but it sounds more like generalized anxiety. I might suggest you google the term "generalized anxiety" or "GAD" or "childhood anxiety" or similar phrases to see if the descriptions are similar to those of your daughter. There are many articles and books in your bookstore, on-line or library with information on anxiety issues.
I'm assuming you have not yet taken your daughter to the children's hospital yet. If GAD is the problem, anxiety disorders are classified under mental health and so the family doctor (assuming the hospital is a competent one) made a good choice. Anxiety is a very common ailment and highly, highly treatable. Anxiety is an inherited trait (just as is the colour of your child's eyes) so there is nothing to be frightened about. Please let us know what the doctor says at the children's hospital so that we can better help you. I wish you the best ...